The expression \(27x^3 - 64y^3\) is a difference of cubes. It can be factored using the formula for the difference of cubes, which is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In this case, we can identify \(a\) and \(b\) as follows:
- \(a = 3x\) (since \( (3x)^3 = 27x^3 \))
- \(b = 4y\) (since \( (4y)^3 = 64y^3 \))
Now, we can apply the difference of cubes formula:
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Compute \(a - b\): \[ 3x - 4y \]
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Compute \(a^2\), \(ab\), and \(b^2\): \[ a^2 = (3x)^2 = 9x^2 \] \[ ab = (3x)(4y) = 12xy \] \[ b^2 = (4y)^2 = 16y^2 \]
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Combine these results: \[ a^2 + ab + b^2 = 9x^2 + 12xy + 16y^2 \]
Now, substituting back into the factored form, we get:
\[ 27x^3 - 64y^3 = (3x - 4y)(9x^2 + 12xy + 16y^2) \]
Thus, the linear factor of the expression \(27x^3 - 64y^3\) is:
\[ 3x - 4y \]